Hyperbola

The normal to the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over 9} = 1\) at the point \(\left( {8,3\sqrt 3 } \right)\) on it passes through the point :

Let the tangent drawn to the parabola \(y^{2}=24 x\) at the point \((\alpha, \beta)\) is perpendicular to the line \(2 x+2 y=5\). Then the normal to the hyperbola \(\frac{x^{2}}{\alpha^{2}}-\frac{y^{2}}{\beta^{2}}=1\) at the point $$(\alpha...

If the line \(x-1=0\) is a directrix of the hyperbola \(k x^{2}-y^{2}=6\), then the hyperbola passes through the point :

Let the eccentricity of the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\) be \({5 \over 4}\). If the equation of the normal at the point \(\left( {{8 \over {\sqrt {5} }},{{12} \over {5}}} \right)\) on the hyperbola is ...

Let the equation of two diameters of a circle \(x^{2}+y^{2}-2 x+2 f y+1=0\) be \(2 p x-y=1\) and \(2 x+p y=4 p\). Then the slope m \(\in\) \((0, \infty)\) of the tangent to the hyperbola \(3 x^{2}-y^{2}=3\) passing through the centre of t...

Let the foci of the ellipse \(\frac{x^{2}}{16}+\frac{y^{2}}{7}=1\) and the hyperbola \(\frac{x^{2}}{144}-\frac{y^{2}}{\alpha}=\frac{1}{25}\) coincide. Then the length of the latus rectum of the hyperbola is :

Let the hyperbola \(H:{{{x^2}} \over {{a^2}}} - {y^2} = 1\) and the ellipse \(E:3{x^2} + 4{y^2} = 12\) be such that the length of latus rectum of H is equal to the length of latus rectum of E. If \({e_H}\) and \({e_E}\) are the eccentriciti...

Let A (sec\(\theta\), 2tan\(\theta\)) and B (sec\(\phi\), 2tan\(\phi\)), where \(\theta\) + \(\phi\) = \(\pi\)/2, be two points on the hyperbola 2x2 \(-\) y2 = 2. If (\(\alpha\), \(\beta\)) is the point of the intersection of the normals to...

The locus of the mid points of the chords of the hyperbola x2 \(-\) y2 = 4, which touch the parabola y2 = 8x, is :

The point \(P\left( { - 2\sqrt 6 ,\sqrt 3 } \right)\) lies on the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\) having eccentricity \({{\sqrt 5 } \over 2}\). If the tangent and normal at P to the hyperbola intersect it...

The locus of the centroid of the triangle formed by any point P on the hyperbola \(16{x^2} - 9{y^2} + 32x + 36y - 164 = 0\), and its foci is :

The locus of the point of intersection of the lines \(\left( {\sqrt 3 } \right)kx + ky - 4\sqrt 3 = 0\) and \(\sqrt 3 x - y - 4\left( {\sqrt 3 } \right)k = 0\) is a conic, whose eccentricity is _________.

A hyperbola passes through the foci of the ellipse \({{{x^2}} \over {25}} + {{{y^2}} \over {16}} = 1\) and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentrici...

Let a line L : 2x + y = k, k > 0 be a tangent to the hyperbola x2 \(-\) y2 = 3. If L is also a tangent to the parabola y2 = \(\alpha\)x, then \(\alpha\) is equal to :

Consider a hyperbola H : x2 \(-\) 2y2 = 4. Let the tangent at a point P(4, \({\sqrt 6 }\)) meet the x-axis at Q and latus rectum at R(x1, y1), x1 > 0. If F is a focus of H which is nearer to the point P, then the area of \(\Delta\)QFR is eq...

The locus of the midpoints of the chord of the circle, x2 + y2 = 25 which is tangent to the hyperbola, \({{{x^2}} \over 9} - {{{y^2}} \over {16}} = 1\) is :

If e1 and e2 are the eccentricities of the ellipse,
\({{{x^2}} \over {18}} + {{{y^2}} \over 4} = 1\) and the hyperbola, \({{{x^2}} \over 9} - {{{y^2}} \over 4} = 1\) respectively and (e1, e2) is a point on the ellipse,
15x2 + 3y2 = k, then ...

If a hyperbola passes through the point
P(10, 16) and it has vertices at (± 6, 0), then the
equation of the normal to it at P is :

If the line y = mx + c is a common tangent to
the hyperbola
\({{{x^2}} \over {100}} - {{{y^2}} \over {64}} = 1\) and the circle
x2
+ y2
= 36, then which one of the following is
true?

Let P(3, 3) be a point on the hyperbola, \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\). If the normal to it at P intersects the x-axis
at (9, 0) and e is its eccentricity, then the ordered pair (a2, e2) is equal to :

A hyperbola having the transverse axis of
length
\(\sqrt 2\) has the same foci as that of the ellipse
3x2 + 4y2 = 12, then this hyperbola does not
pass through which of the following points?

Let e1
and e2
be the eccentricities of the
ellipse, \({{{x^2}} \over {25}} + {{{y^2}} \over {{b^2}}} = 1\)(b < 5) and the hyperbola,
\({{{x^2}} \over {16}} - {{{y^2}} \over {{b^2}}} = 1\) respectively satisfying e1e2
= 1. If \(\alpha\)
...

A line parallel to the straight line 2x – y = 0 is
tangent to the hyperbola
\({{{x^2}} \over 4} - {{{y^2}} \over 2} = 1\) at the point
\(\left( {{x_1},{y_1}} \right)\). Then \(x_1^2 + 5y_1^2\) is equal to :

For some \(\theta \in \left( {0,{\pi \over 2}} \right)\), if the eccentricity of the
hyperbola, x2–y2sec2\(\theta\) = 10 is
\(\sqrt 5\) times the
eccentricity of the ellipse, x2sec2\(\theta\) + y2 = 5, then
the length of the latus rect...

Let \(0 < \theta < {\pi \over 2}\). If the eccentricity of the
hyperbola \({{{x^2}} \over {{{\cos }^2}\theta }} - {{{y^2}} \over {{{\sin }^2}\theta }}\) = 1 is greater
than 2, then the length of
its latus rectum lies in the interval :

A hyperbola has its centre at the origin, passes through the point (4, 2) and has transverse axis of length 4 along the x-axis. Then the eccentricity of the hyperbola is :

If the line y = mx + 7\(\sqrt 3\) is normal to the
hyperbola
\({{{x^2}} \over {24}} - {{{y^2}} \over {18}} = 1\) , then a value of m is :

If the eccentricity of the standard hyperbola
passing through the point (4,6) is 2, then the
equation of the tangent to the hyperbola at (4,6)
is :

If the vertices of a hyperbola be at (–2, 0) and (2, 0) and one of its foci be at (–3, 0), then which one of the following points does not lie on this hyperbola?

Equation of a common tangent to the parabola y2 = 4x and the hyperbola xy = 2 is :

If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13, then the
eccentricity of the hyperbola is :

A circle cuts a chord of length 4a on the x-axis and passes through a point on the y-axis, distant 2b from the origin. Then the locus of the centre of this circle, is :

The equation of a tangent to the hyperbola 4x2 – 5y2 = 20 parallel to the line x – y = 2 is :

If a directrix of a hyperbola centred at the origin and passing through the point (4, –2\(\sqrt 3\) ) is 5x = 4\(\sqrt 5\) and
its eccentricity is e, then :

If 5x + 9 = 0 is the directrix of the hyperbola 16x2
– 9y2
= 144, then its corresponding focus is :

The locus of the point of intersection of the lines, \(\sqrt 2 x - y + 4\sqrt 2 k = 0\) and \(\sqrt 2 k\,x + k\,y - 4\sqrt 2 = 0\) (k is any non-zero real parameter), is :

If the tangents drawn to the hyperbola 4y2 = x2 + 1 intersect the co-ordinate axes at the distinct points A and B then the locus of the mid point of AB is :

A normal to the hyperbola, 4x2 \(-\) 9y2 = 36 meets the co-ordinate axes \(x\) and y at A and B, respectively. If the parallelogram OABP (O being the origin) is formed, then the ocus of P is :

Tangents are drawn to the hyperbola 4x2 - y2 = 36 at the points P and Q.
If these tangents intersect at the
point T(0, 3) then the area (in sq. units) of \(\Delta\)PTQ is :

The locus of the point of intersection of the straight lines,
tx \(-\) 2y \(-\) 3t = 0
x \(-\) 2ty + 3 = 0 (t \(\in\) R), is :

A hyperbola passes through the point P\(\left( {\sqrt 2 ,\sqrt 3 } \right)\) and has foci at \(\left( { \pm 2,0} \right)\). Then the tangent to this hyperbola at P also passes through the point :

Let a and b respectively be the semitransverse and semi-conjugate axes of a
hyperbola whose eccentricity satisfies the equation 9e2 − 18e + 5 = 0. If S(5, 0) is a focus and 5x = 9 is the corresponding directrix of this hyperbola, then a2 − ...

A hyperbola whose transverse axis is along the major axis of the conic, \({{{x^2}} \over 3} + {{{y^2}} \over 4} = 4\) and has vertices at the foci of this conic. If the eccentricity of the hyperbola is \({3 \over 2},\) then which of the fo...

The eccentricity of the hyperbola whose length of the latus rectum is equal to \(8\) and the length of its conjugate axis is equal to half of the distance between its foci, is :

The normal to a curve at \(P(x,y)\) meets the \(x\)-axis at \(G\). If the distance of \(G\) from the origin is twice the abscissa of \(P\), then the curve is a :

For the Hyperbola \({{{x^2}} \over {{{\cos }^2}\alpha }} - {{{y^2}} \over {{{\sin }^2}\alpha }} = 1\) , which of the following remains constant when \(\alpha\) varies\(=\)?

The locus of a point \(P\left( {\alpha ,\beta } \right)\) moving under the condition that the line \(y = \alpha x + \beta\) is tangent to the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\) is :

The foci of the ellipse \({{{x^2}} \over {16}} + {{{y^2}} \over {{b^2}}} = 1\) and the hyperbola \({{{x^2}} \over {144}} - {{{y^2}} \over {81}} = {1 \over {25}}\) coincide. Then the value of \({b^2}\) is :